Block Stability for MAP Inference
This work provides a theoretical explanation for the empirical success of approximate MAP inference in computer vision, addressing a gap in understanding why existing algorithms perform well despite strong stability assumptions not holding in practice.
The paper tackles the problem of understanding why approximate MAP inference works well in practice by introducing a relaxed block stability condition, which requires only portions of an input instance to be stable, and proves that under this condition, the pairwise LP relaxation is persistent on stable blocks, with empirical evaluation showing large stable regions in real-world computer vision instances.
To understand the empirical success of approximate MAP inference, recent work (Lang et al., 2018) has shown that some popular approximation algorithms perform very well when the input instance is stable. The simplest stability condition assumes that the MAP solution does not change at all when some of the pairwise potentials are (adversarially) perturbed. Unfortunately, this strong condition does not seem to be satisfied in practice. In this paper, we introduce a significantly more relaxed condition that only requires blocks (portions) of an input instance to be stable. Under this block stability condition, we prove that the pairwise LP relaxation is persistent on the stable blocks. We complement our theoretical results with an empirical evaluation of real-world MAP inference instances from computer vision. We design an algorithm to find stable blocks, and find that these real instances have large stable regions. Our work gives a theoretical explanation for the widespread empirical phenomenon of persistency for this LP relaxation.